Nonlinear anisotropic degenerate parabolic-hyperbolic equations with stochastic forcing

We are concerned with nonlinear anisotropic degenerate parabolic-hyperbolic equations with stochastic forcing, which are heterogeneous ( i.e., not space-translational invariant). A unified framework is established for the continuous dependence estimates, fractional BV regularity estimates, and well-posedness for stochastic kinetic solutions of
the nonlinear stochastic degenerate parabolic-hyperbolic equation.
In particular, we establish the well-posedness of the nonlinear stochastic equation in $L^p \cap N^{\kappa,1}$ for $p\in [1,\infty)$ and the $\kappa$--Nikolskii space $N^{\kappa,1}$ with $\kappa\in (0,1]$,
and the $L^1$--continuous dependence of the stochastic kinetic solutions not only on the initial data, but also on the degenerate diffusion matrix function, the flux function, and the multiplicative noise function involved in the nonlinear equation.